Problem $2(30$ Points$)$

A microscopic spring$-$mass system has a mass $\backslash ($ m$=5\backslash$times $10^\{-26\}\backslash$mathrm$\{$~kg$\}\backslash )$ and the energy gap between the $2$nd and $3$rd excited states is $5$ eV $.$

a$)(5$ Points$)$ Calculate the energy gap between the $4$ th and $7$th excited states: $\backslash (\backslash$Delta$\_\{4\mathrm{.}7\}=\backslash )[$eV$]$

b$)(5$ Points$)$ The energy of the ground state is: $\backslash (\backslash$mathbf$\{$E$\}\_\{0\}=\backslash )[$eV$]$

c$)(5$ Points$)$ The stiffness of the spring is: $\backslash ($ k$\_\{3\}=\backslash )[\backslash (\backslash$mathrm$\{$N$\}/\backslash$mathrm$\{$m$\}\backslash )]$

d$)(5$ Points$)$ What is the wavelength of the smallest energy photon emitted by this system? $\backslash (\backslash$mathrm$\{$A$\}=\backslash )[$m$]$

e$)(3$ Points$)$ If the stiffness of the spring increases, the wavelength calculated in the previous part

f$)(2$ Points$)$ If the mass increases, the energy gap between successive energy levels

g$)(5$ Points$)$ What should the stiffness of the spring be$,$ so that the transition from the $3$rd excited state to the $2$ nd excited state emits a photon with energy $5\mathrm{.}5$ eV $?\backslash (\backslash$mathrm$\{$k$\}\_\{\backslash$mathrm$\{$s$\}\}=\backslash )[$N$/$m$]$